Interference between independent fluctuating condensates
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Edited by Paul C. Martin, Harvard University, Cambridge, MA, and approved January 31, 2006 (received for review December 2, 2005)
Abstract
We consider a problem of interference between two independent condensates that lack true longrange order. We show that their interference pattern contains information about correlation functions within each condensate. As an example, we analyze the interference between a pair of onedimensional interacting Bose liquids. We find universal scaling of the average fringe contrast with system size and temperature that depends only on the Luttinger parameter. Moreover, the full distribution of the fringe contrast, which is also equivalent to the full counting statistics of the interfering atoms, changes with interaction strength and lends information on highorder correlation functions. We also demonstrate that the interference between twodimensional condensates at finite temperature can be used as a direct probe of the Kosterlitz–Thouless transition. Finally, we discuss the generalization of our results to describe the interference of a periodic array of independent fluctuating condensates.
An important property of Bose–Einstein condensates is the existence of a coherent macroscopic phase. Thus, a crucial benchmark in the study of such systems was the observation of interference fringes when two independent condensates were allowed to expand and overlap (1). This “twoslit” experiment was carried out with cold atoms in threedimensional harmonic traps, where a true condensate exists. The interference fringe amplitude should then be proportional to the condensate fraction, as was indeed observed. However, with current trapping technology it is possible to confine the bosonic atoms to one (2–4) or two dimensions (5), where a true condensate may not exist. Instead, these systems are characterized by offdiagonal correlations that either decay as a powerlaw or decay exponentially in space. What is the interference pattern that arises when two such imperfect condensates are allowed to expand and overlap? This question is not just of general academic interest. Recently there have been a number of experiments showing the interference between independent condensates (see, for example, refs. 6–8).
Here we address this problem theoretically and show that the result depends crucially on the correlations within each condensate. Therefore, such an experiment would provide a direct and simple probe of the spatial phase correlations. In principle, spatial phase correlations may also be extracted from juggling experiments (9–11) or the momentum distribution measured by the free expansion of a single condensate (2). However, creating strongly interacting lowdimensional systems typically requires using lowdensity atomic gases, which makes juggling experiments very challenging. In addition, the highly anisotropic expansion of lowdimensional condensates inhibit measurements of the momentum distribution in the slowly expanding longitudinal direction. A method for probing the phase correlations directly in real space would therefore be very useful.
Results
The simplest geometry that we consider is illustrated in Fig. 1and consists of two parallel onedimensional condensates a distance, d, apart. After the atoms are released from the trap, they are allowed to expand to a transverse size much larger than d, although no significant expansion occurs in the axial direction. An absorption image is then taken by a probe beam directed along the condensate axis. A similar setup is considered for twodimensional condensates on parallel planes (see Fig. 2). As usual, the absorption image gives the instantaneous threedimensional density profile integrated along the beam axis. ρ(x) = ∫_{0} ^{L} dz a _{tof} ^{†} (x, z) a _{tof} (x, z), where a _{tof} ^{†} are the Bose creation operators with the subscript “time of flight” (tof), emphasizing that the corresponding operators are taken after free expansion of atoms, z is the axial coordinate and x is the coordinate along the detector (see Fig. 1). The length L is typically given by the focal length of the imaging beam. It may also be controlled more precisely by applying magnetic field gradients so that only a specified section of the cloud is resonant with the probe light. In principle, one can consider an experiment with a probe beam orthogonal to the plane containing two parallel onedimensional condensates. In this case, it is possible to integrate the resulting interference image within an arbitrary interval and obtain dependence of the interference contrast on L (note that this dependence characterizes a single run and a series of experiments is still needed to find the average contrast). The other advantage of this setup is that it can reveal the presence of dipolar oscillations in individual condensates. These modes correspond to the center of mass motion and are not affected by interactions. Dipolar oscillations induce an overall tilt in the interference peak position and can be easily removed by integrating ρ(x) along a line tilted with respect to the z axis. However, because most of the current experimental systems do not allow imaging beams that are perpendicular to onedimensional condensates, we concentrate on the setup shown in Fig. 1.
To discuss the interference contrast, we consider the correlation function of the density operator, where r _{i} stands for (x _{i} , z _{i} ). Singleparticle operators in Eq. 1 should be taken after the expansion time t. We can relate them to operators before the expansion (12): a _{tof}(x, z) = a _{1}(z)e^{iQ1(x)x−iQ12t/2m} + a _{2} (z)e^{iQ2(x)x−iQ12t/2m}, with a _{1} and a _{2} being operators in the two condensates and Q _{1} and Q _{2} = m(x ± d/2)/ℏt. We therefore find that the correlation function in Eq. 1 has an oscillating component at wave vector Q = md/ℏt. Here A _{Q} = ∫dza _{1} ^{†}(z)a _{2} (z) is the quantum observable corresponding to the amplitude of the interference fringes. It can be extracted from the TOF absorption image by taking the Fourier transform of the density profile. Alternatively, one can directly probe A _{Q} ^{2} by studying the oscillating component in the density autocorrelation function. Both methods were successfully used in recent experiments (13, 14). In practice, it might be easier to study the interference contrast rather than the absolute value of the fringe amplitude. In this case, one has to divide A _{Q} by the imaging length L. If the two condensates are decoupled from each other, the expectation value of 〈A _{Q} 〉 vanishes, which does not mean that A _{Q}  is zero in each individual measurement, but it does show that the phase of A _{Q} is random (15). Said differently, A _{Q} is finite in each experimental run, but its average over many experiments vanishes. To determine the amplitude of interference fringes in individual measurements, one should consider an expectation value of the quantity that does not involve the random phase of A _{Q} . This consideration naturally brings us to Eq. 3 . From shot to shot, A _{Q}  ^{2} fluctuates as well, and Eq. 3 gives its average value.
If the two condensates are identical (but still independent), we may simplify Eq. 3 :
Here, we neglected boundary effects by integrating over the center of mass coordinate and assuming that the correlations depend only on (z _{1} − z _{2} ). Eq. 4 can be generalized for the case of parallel twodimensional condensates by taking z to represent the planar coordinates.
To gain intuition into the physical meaning of the fringe amplitude, let us first address two simple limiting cases. First, consider the situation where 〈a
^{†}
(z)a(0)〉 decays exponentially with distance with a correlation length ξ ≪ L. Then, Eq. 4
implies that A
_{Q}
 ∝
OneDimensional Bose Liquids.
We proceed to discuss the case of a onedimensional interacting gas. We first consider a system at sufficiently low temperature, ξ
_{T}
≫ L, where ξ
_{T}
is the temperaturedependent correlation length defined in Eqs. 17
and
18
. In this regime, the correlations decay as a powerlaw rather than decay exponentially. We therefore, expect that the fringe amplitude will somehow interpolate between the two simple limits considered above. Specifically, at long wavelengths, the onedimensional Bose gas is described by a Luttinger liquid (16), and the longdistance offdiagonal correlations behave as
Here, ρ is the particle density, ξ
_{h}
is the healing length, which also serves as the shortrange cutoff, and K is the Luttinger parameter. For bosons with a repulsive shortrange potential, K ranges between 1 and ∞, with K = 1 corresponding to strong interactions, or “impenetrable” bosons, and K → ∞ corresponding to noninteracting bosons. Substituting Eq. 5
into Eq. 4
and assuming that L ≫ ξ
_{h}
, we arrive at one of our main results:
where C is a constant of order unity. Thus, we see that the amplitude of the interference fringes (Ā
_{Q}
≡
Having derived the amplitude of interference fringes, an interesting question is how this amplitude fluctuates from one experimental run to the next. To answer this question, one should consider higher moments of the operator A
_{Q}

^{2}
. We find that all moments have the general form: 〈A
_{Q}

^{2n}
〉 = 〈A
_{Q}

^{2}
〉
^{n}
F
_{n}
(K). It follows that when A
_{Q}

^{2}
is normalized, its distribution function P(A
_{Q}

^{2}
/〈A
_{Q}

^{2}
〉) is fully determined by the Luttinger parameter, K. In particular, for large K values, the function P(A
_{Q}

^{2}
/〈A
_{Q}

^{2}
〉) becomes very narrow, characterized by the width σ ≡
We showed above that it is possible to extract K by analyzing the scaling of the fringe amplitude with system size or by analyzing its distribution at a given system size. Another approach involves changing the angle θ between the probe beam and the condensate axis while keeping the imaging length fixed. The resulting absorption image then corresponds to integration of the cloud density along a line at an angle θ to the z axis. Then Eq. 1 should be changed accordingly. Let x̄ be the transverse coordinate on the screen (z = 0), then, in the second term in the righthand side of Eq. 1 , we substitute x = x̄ − z tan θ. Then, we obtain the analogue of Eq. 4 : where q(θ) = k _{0} tan θ. For sufficiently large imaging length (qL ≫ 1), Eq. 7 yields Thus, the Luttinger parameter may be extracted from the angle dependence of the fringe amplitude. For qL ≫ 1 (i.e., very small angles), Eq. 7 reduces to Eq. 4 . Note that if one uses the imaging beam orthogonal to the condensates, then θ will simply be the angle between the z axis and the direction of integration of the interference contrast.
Before concluding this section, let us address the effect of temperature. It is well known that, at any finite temperature, the offdiagonal correlations in a onedimensional Bose system must be shortranged. Specifically, at sufficiently long distances, offdiagonal correlations decay exponentially with a correlation length ξ
_{T}
∼ 1/T. The zerotemperature results presented above are valid at sufficiently low temperature that ξ
_{T}
≫ L. At higher temperature such that ξ
_{T}
≪ L, the scaling of the fringe amplitude with length must be Ā_{Q}
 ∼
TwoDimensional Systems.
We now consider a pair of parallel twodimensional condensates. In direct analogy to the onedimensional condensates, the imaging axis may be taken parallel or at some angle to the plane of the condensates. In the former case, one should consider the scaling of the fringe amplitude with imaging length, whereas, in the latter case, one should consider the variation with angle.
It is well known that in two dimensions longrange order may exist only at zero temperature. At sufficiently low temperatures, offdiagonal correlations are algebraic, with for r ≫ ξ _{h} . In contrast, above the Kosterlitz–Thouless (KT) transition at T = T _{c} , the correlations decay exponentially. We will show that this transition is characterized by a jump in the behavior of the fringe amplitude, related to the well known universal jump of the superfluid stiffness at T _{c} .
The exponent in Eq. 10 is given by α = mT/2πρ _{s} (T)ℏ ^{2} . For weakly interacting bosons at temperatures well below T _{c} , ρ _{s} (T) is simply equal to the density, ρ. As one approaches the transition, ρ _{s} is renormalized by fluctuations, and, at the transition, ρ_{s} (T _{c} ) = 2mT _{c} /π ℏ ^{2} . Therefore, the exponent α assumes a universal value α _{c} = 1/4 at the transition. Thus, for temperatures T < T _{c} , we have 0 < α < α _{c} .
Let us now discuss the consequences of this physics to the experimentally measurable fringe amplitude. As illustrated in Fig. 1, the interference pattern is now truly twodimensional in the sense that cuts along x at different coordinate, y, display a different fringe pattern. To obtain a onedimensional pattern as a function of x alone, we may integrate the image intensity over an “integration length,” L
_{y}
. Recall that, in addition, the imaging process automatically integrates over an imaging length L
_{z}
along the z axis. Now the generalization of Eq. 4
to the twodimensional case is straightforward:
For simplicity, we assume that L
_{y}
and L
_{z}
are scaled simultaneously as L
_{y}
= L
_{z}
=
The analysis for imaging the twodimensional condensates with a slanted probe beam can be carried over from the onedimensional case. The scaling of the interference contrast with q = k _{0} tan θ, at constant imaging area, is then 〈A _{Q}  ^{2〉} ∼ 1/q ^{2−2α} below the KT transition, and 〈A_{Q}  ^{2} 〉 ∼ 1/(1 + q ^{2}ξ ^{2} ) ^{3/2} above it. Again the transition is characterized by a universal jump of the power at small q. We emphasize that θ can be either the angle between the beam and the z axis (see Fig. 1) or the angle between the y axis and the direction of integration. The latter is preferable, because, within a single experimental shot, it is possible to obtain the whole angular dependence of A _{Q} ^{2} .
Regardless of the experimental approach of choice, the interference between a parallel pair of independent twodimensional condensates can serve as a direct probe of KT physics. However a word of caution is in order. The correlation length, which coincides with the healing length at very low temperatures (19) (ξ
_{T}
≈ ξ_{h} = ℏ/
Discussion
We considered a pair of interfering quasicondensates; however, most of our arguments can be generalized to the case of several independent condensates. Of particular interest is a periodic array of tubes (2–4) or pancakes created by an optical potential (5, 21, 22). The interference pattern in this case shows correlations at a set of wave vectors Q _{n} = nQ, where n is an integer and Q is determined by the distance between neighboring condensates. The size and angle dependence of the average interference amplitudes for each of these wave vectors should have the same scaling properties as two quasicondensates. However, the distribution function of fringe amplitudes will be different. In particular, in the limit of a large number of condensates, the distribution function should become very narrow. This result follows immediately from the observation that in this limit, higherorder correlation functions in TOF images are dominated by products of two point correlation function in different condensates, so there should be no broadening associated with Eq. 15 below.
Another point worth making regards the possibility of making analogous experiments with Fermions. For example, one can consider an interference of two independent onedimensional fermionic systems. One obvious difference from the bosonic case will be the change of sign in the correlation function (see Eq. 2
), reflecting different statistics of the fermions (this corresponds to fermion antibunching). More importantly, the correlation function decays as 1/x
^{1/2(K+1/K)}
, i.e., as 1/x or faster. This scaling means that the integral in Eq. 4
is dominated by short distances, at which the Luttinger liquid description is not sufficient, and that the integral converges as L → ∞. Infrared convergence of Eq. 4
implies trivial scaling A
_{Q}
 ∝
In conclusion, we analyzed the interference between two independent quasicondensates. We showed that scaling properties of interference fringes directly probe the algebraic offdiagonal correlations. In particular, for onedimensional condensates, the scaling with imaging length or with temperature allows the extraction of the Luttinger parameter. In the case of twodimensional condensates, this method provides a unique probe of the KT transition. We also argued that, in the onedimensional case, one can use the distribution function of the interference amplitude (which is also equivalent to the full counting statistics of interfering bosons) as the qualitative probe of the Luttinger constant. In particular, at K ≫ 1 the distribution is narrow and at K → 1 or, at finite temperatures, it becomes wide Poissonian (see Fig. 2). In the twodimensional case, we expect a sharp change in the shape of the distribution function at the KT transition. The scaling analysis remains intact if more than two independent condensates are present, but the distribution functions can no longer be used as a probe of the correlations.
Methods
Luttinger Liquid Parameter.
The Luttinger liquid provides a universal longwavelength description of onedimensional, interacting Bose liquids that allows the calculation of the longdistance behavior of correlations such as Eq. 5 . In certain regimes, it is possible to derive the Luttinger parameter, K, and the healing length, ξ _{h} , from the microscopic interactions. In particular, for bosons with weak contact interactions, relevant for ultracold atom systems discussed in this work, one can use Bogoliubov theory to obtain (23, 24) Here, γ ≡ 2/(a _{1d} _{ρ} ) ≲ 1 is a dimensionless measure of the interaction strength, and a _{1d} is the onedimensional scattering length. Analytic expressions for these parameters may also be derived in the limit of strong contact interaction γ ≫ 1 (23):
Moments of the Fringe Amplitude.
All of the moments in the distribution of A _{Q}  ^{2} can be obtained by generalizing the twopoint correlation function in Eq. 3 to the 2npoint correlation function For bosons with repulsive interactions described by the Luttinger parameter, K, we have which is precisely of the form 〈A _{Q}  ^{2n} 〉 = 〈A _{Q}  ^{2} 〉 ^{n} F _{n} (K). Integrals of the type appearing in Eq. 16 have been discussed by Fendley et al. (25), who demonstrated that they can be calculated by using special properties of Jack polynomials. From the knowledge of all moments, one can, in principle, construct the full distribution of the interference fringes amplitude. In this paper, we only discuss the limits of weak (K ≫ 1) and strong (K ≈ 1) interactions.
Finite Temperature Correlations in One Dimension.
The finitetemperature, offdiagonal correlations are given by ref. 23: where the thermal correlation length, ξ _{T} , is Eq. 17 is valid for sufficiently low temperatures so that ξ _{T} ≫ ξ _{h} or, equivalently, T ≪ ℏ ^{2} /mξ _{h} ^{2}. For z ≪ ξ _{T} , Eq. 17 reduces to the zero temperature correlation (5). In the opposite limit, z ≫ ξ _{T} , the correlation function given by Eq. 17 may be approximated by As we already noted, for sufficiently low temperatures when ξ _{T} > L, the fringe amplitude may be found from Eq. 6 . For L ≫ ξ _{T} , Eq. 19 implies Finally, substituting Eq. 18 for ξ _{T} gives Eq. 9 .
We also note that the angular dependence of the fringe amplitude at finite temperature is given by From this expression, it is harder to extract K directly because of uncertainty in the determination of ξ _{h} and, hence, ξ _{T} .
Acknowledgments
We thank P. Fendley, M. Greiner, V. Gritsev, Z. Hadzibabic, M. Lukin, M. Oberthaller, M. Oshikawa, J. Schmiedmayer, V. Vuletic, D. Weiss, and K. Yang for useful discussions. This work was supported in part by the U.S.–Israel Binational Science Foundation and by National Science Foundation Grant DMR0132874.
Footnotes
 ^{§}To whom correspondence should be addressed. Email: asp{at}bu.edu

Author contributions: A.P., E.A., and E.D. designed research, performed research, and wrote the paper.

Conflict of interest statement: No conflicts declared.

This paper was submitted directly (Track II) to the PNAS office.
 Abbreviations:
 KT,
 Kosterlitz–Thouless.
Abbreviation:
 © 2006 by The National Academy of Sciences of the USA
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